If I buy one will it work from Tucson to New York
It will work from anywhere in the world as long you are connected on internet from both ends ( the camera and phone, tablet, or pc)
Is there othe expense to use this, monthly fees? Will it work 20 miles away?
There are no monthly fees, it will work from anywhere in the world as long you are connected to the internet.
Can I mount this on the wall and does it come with the backup battery in it case electric goes out
This camera comes with a wall mounting bracket. Sorry, does not have a backup battery.
Are they for outdoor use also
Yes, they can be used outdoor but need to be protected from the direct exposure of rain and snow, they are not waterproof.
Is the camera whether resistant?
You can Install it under a canopy, or your porch where the camera will be protected from rain.
If you need waterproof camera check this one from the link bellow